Gaussianity and Simulability of Cliffords and Matchgates

TL;DR

This paper explores the combined class of Clifford and matchgate circuits, extending their known simulability and revealing their shared Gaussian structure, thus broadening understanding of their computational properties.

Contribution

It generalizes the simulability of Clifford-matchgate hybrid circuits and introduces a Gaussian framework unifying both circuit families.

Findings

Hybrid circuits are more broadly simulable than previously known.

Pauli expectation values can be efficiently computed for these hybrids.

Both Clifford and matchgate circuits can be understood as Gaussian processes.

Abstract

Though Cliffords and matchgates are both examples of classically simulable circuits, they are considered simulable for different reasons. The celebrated Gottesman-Knill explains the simulability Cliffords, and the efficient simulability of matchgates is understood via Pfaffians of antisymmetric matrices. We take the perspective that by studying Clifford-matchgate hybrid circuits, we expand the set of known simulable circuits and reach a better understanding of what unifies these two circuit families. While the simulability of Clifford conjugated matchgate circuits for single qubit outputs has been briefly considered, the simulability of Clifford and matchgate hybrid circuits has not been generalized up to this point. In this paper we extend that work, studying simulability of marginals as well as Pauli expectation values of Clifford and matchgate hybrid circuits. We describe a hierarchy of Clifford circuits, and find that as we consider more general Cliffords, we lose some amount of simulability of bitstring outputs. We then show that the known simulability of Pauli expectation values of Clifford circuits acting on product states can be generalized to Clifford circuits acting after any matchgate circuit. We conclude with general discussion about the relationship between Cliffords and matchgates, and show that both circuit families can be understood as being Gaussian.